Rebecca Maloy’s Inside the Offertory: Aspects of Chronology and Transmission is mainly about Gregorian chant in the offertory, but early on she summarizes current opinion regarding the origins of the offertory. Contrary to some earlier liturgical historians, “A lay offering during the liturgy of the early church . . . cannot be substantiated.”
She elaborates the evidence: “While Early Christian writers do admonish the laity to offer gifts, their offering probably took place before the service. The Testamentum domini, a Syrian document perhaps from the fifth century, directs the deacons to receive offerings in a room on the right side of the church set aside for that purpose, clearly indicating that the offering was not part of the service. Jerome refers to the custom of deacons announcing the names of those who had offered the bread and wine during Mass. A similar tradition is witnessed in a letter of Pope Innocent I, who recommends that these prayers be incorporated into the Eucharistic rite.” But, “if the laity offered during the service, there would be no need to announce their names.”
The earliest unquestionable evidence comes from fourth-century North Africa:
“Augustine’s writings do suggest that the laity approached the altar to present their offerings. He expresses regret, for example, that a woman captured by barbarians could not make her offering at the altar.” It is not clear, Maloy argues, whether Augustine’s reference to singing of hymns ante oblationem is a reference to “offertory chants” (17).
For Rome, the evidence is even later: “The offertory rite in the papal liturgy makes its first appearance in Ordo Romanus I, which describes the liturgy celebrated in the various stational churches around 700-750″ (21). According to the Old Gelasian sacramentary (cf. 628-715), “an offering by the ordinary laity [was made] in the context of urban liturgy.”